Teichm\"uller and lamination spaces with pinnings

TL;DR

This paper explores the structure of Teichm"uller and lamination spaces with pinnings, providing formulas and topological descriptions, and investigates duality maps and basis amalgamation in these moduli spaces.

Contribution

It offers a new topological perspective on $ ext{P}$-laminations, relates functions on Teichm"uller spaces with pinnings, and studies duality maps and basis amalgamation in these spaces.

Findings

Formulas relating $ ext{P}$-laminations and functions on Teichm"uller space.

Topological description of the tropicalized amalgamation map.

Analysis of compatibility of Fock--Goncharov duality maps.

Abstract

We describe the spaces of the positive and tropical points of the moduli space $\mathcal{P}_{PGL_2,\Sigma}$ introduced by Goncharov--Shen [GS19] as certain Teichm\"uller and lamination spaces, respectively, with additional data of pinnings. In the case where the surface $\Sigma$ has no punctures, we obtain the formulae relating various functions on the Teichm\"uller space with pinnings: $\lambda$-lengths, cross ratio coordinates, and Wilson lines. A topological description of the tropicalized amalgamation map is given in terms of $\mathcal{P}$-laminations. Based on our topological study of these "$\mathcal{P}$-type" spaces, we investigate the compatibility of the Fock--Goncharov duality maps $\mathbb{I}_\mathcal{A}$, $\mathbb{I}_\mathcal{X}$ constructed by [FG06,FG07,MSW13,GS15] under the extended ensemble map. We also discuss the amalgamation of bracelet bases.